[Paper Review] Topological Non-Hermitian skin effect

Rijia Lin, Tommy Tai+3 more|arXiv (Cornell University)|Feb 6, 2023

Quantum Mechanics and Non-Hermitian Physics4 citations

TL;DR

This paper provides a comprehensive review of the non-Hermitian skin effect (NHSE), focusing on its interplay with topology in non-Hermitian systems. It introduces modified bulk-boundary correspondence, spectral winding topology, and hybrid skin-topological states, and demonstrates how NHSE emerges in higher dimensions, with nonlinearity, many-body interactions, and in diverse platforms such as photonics, ultracold atoms, and mechanical systems. The key contribution is a unified framework linking non-Hermitian topology, spectral topology on the complex plane, and experimental realizations across multiple physical systems.

ABSTRACT

This article reviews recent developments in the non-Hermitian skin effect (NHSE), particularly on its rich interplay with topology. The review starts off with a pedagogical introduction on the modified bulk-boundary correspondence, the synergy and hybridization of NHSE and band topology in higher dimensions, as well as, the associated topology on the complex energy plane such as spectral winding topology and spectral graph topology. Following which, emerging topics are introduced such as non-Hermitian criticality, dynamical NHSE phenomena, and the manifestation of NHSE beyond the traditional linear non-interacting crystal lattices, particularly its interplay with quantum many-body interactions. Finally, we survey the recent demonstrations and experimental proposals of NHSE.

Motivation & Objective

• To establish a unified theoretical framework for the non-Hermitian skin effect (NHSE) and its interplay with topological invariants in non-Hermitian systems.
• To resolve the breakdown of conventional bulk-boundary correspondence in non-Hermitian systems by introducing non-Bloch band theory and generalized Brillouin zone (GBZ) concepts.
• To explore the emergence of higher-order NHSE, hybrid skin-topological modes, and spectral topology on the complex energy plane.
• To investigate NHSE in non-linear, interacting, and disordered systems beyond standard lattice models.
• To survey and evaluate experimental realizations of NHSE across photonic, electronic, ultracold atomic, mechanical, and active media platforms.

Proposed method

• Develops non-Bloch band theory to describe the modified bulk-boundary correspondence in non-Hermitian systems, replacing the conventional Brillouin zone with a generalized one (GBZ).
• Introduces spectral winding topology and complex band evolution as topological invariants on the complex energy plane, using the characteristic polynomial of the Hamiltonian.
• Applies an electrostatics analogy to solve the NHSE problem by modeling eigenstate localization as charge distributions on the complex plane.
• Analyzes higher-dimensional systems, including Chern lattices, Weyl semimetals, and higher-order topological insulators, to identify novel skin-topological hybrid states.
• Investigates non-Hermitian criticality and dynamical NHSE in time-periodic (Floquet) systems, including non-divergent asymptotic dynamics.
• Proposes and reviews experimental platforms such as photonic lattices, quantum circuits, mechanical resonators, ultracold atoms, and active networks to realize NHSE.

Experimental results

Research questions

• RQ1 How does the non-Hermitian skin effect modify the conventional bulk-boundary correspondence in topological systems?
• RQ2 What topological invariants govern the NHSE, and how do they differ from those in Hermitian systems?
• RQ3 How does the NHSE manifest in higher dimensions, and what are the signatures of hybrid skin-topological modes?
• RQ4 What role does spectral winding topology on the complex energy plane play in characterizing non-Hermitian systems?
• RQ5 Can the NHSE be realized and probed in non-lattice, non-linear, or many-body quantum systems, and what are the observable signatures?

Key findings

• The non-Hermitian skin effect breaks conventional bulk-boundary correspondence, leading to edge states that cannot be predicted by standard topological invariants under periodic boundary conditions.
• Non-Bloch band theory with a generalized Brillouin zone (GBZ) successfully explains the localization of eigenstates and enables topological characterization without relying on the GBZ.
• Spectral winding topology on the complex energy plane provides a robust topological invariant that characterizes the NHSE, even in systems with complex eigenenergies.
• Higher-order NHSE is observed in systems like Chern lattices and honeycomb lattices, leading to corner or hinge states that are robust against disorder.
• The NHSE can be experimentally realized in photonic, electronic, mechanical, and ultracold atomic systems, with signatures detectable via transport, impedance, or density-of-states measurements.
• Non-Hermitian criticality and exceptional bound states emerge in interacting and non-linear systems, indicating new universality classes beyond conventional Hermitian criticality.